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In 1980, Albert Fathi asked whether the group of area-preservinghomeomorphisms of the 2-disc that are the identity near the boundary is asimple group. In this paper, we show that the simplicity of this group isequivalent to the following fragmentation property in the group of compactlysupported, area preserving diffeomorphisms of the plane: there exists aconstant m such that every element supported on a disc D is the product of atmost m elements supported on topological discs whose area are half the area ofD.

Simplicity of the group of compactly supported area preserving homeomorphisms of the open disc and fragmentation of symplectic diffeomorphisms